Schrödinger operators with strongly attractive graph-type...

Schrödinger operators with stronglyattractive graph-type interaction

Pavel Exner

in collaboration with Kazushi Yoshitomi, Sylwia Kondej,

Francois Bentosela, Pierre Duclos and Milos Tater

exner@ujf.cas.cz

Department of Theoretical Physics, NPI, Czech Academy of Sciences

and Doppler Institute, Czech Technical University

Okayama University, March 24, 2004 – p.1/50

Talk overview

Leaky quantum graphs – why are they interesting?

Schrödinger operators to be considered,Hα,Γ = −∆− αδ(x− Γ)

Geometrically induced discrete spectrum

Punctured manifolds: a perturbation theory

Strong-coupling asymptotics for a compact Γ

Technique: bracketing + coordinate transformation

Infinite manifolds

Periodic case, magnetic field, absolute continuity

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Talk overview

Infinite manifolds

Open questions

Leaky graphs – why are they interesting?

Recall the “standard” graph models:

r r r r Hamiltonian: − ∂2

+ v(xj)

on graph edges,boundary conditions at vertices

Also, generalized graphs – nanotubes + fullerenes, etc.

the edges same above,

−∆LB + v(x) on the manifoldboundary conditions at vertices

Leaky graphs – why are they interesting?

Recall the “standard” graph models:

r r r r Hamiltonian: − ∂2

+ v(xj)

on graph edges,boundary conditions at vertices

Also, generalized graphs – nanotubes + fullerenes, etc.

the edges same above,

−∆LB + v(x) on the manifoldboundary conditions at vertices

Drawbacks of these models

Presence of ad hoc parameters in the b.c. describingbranchings. A natural remedy: use a zero-width limit ina more realistic description

r−→

However, the answer is known so far only forNeumann-type situations [Rubinstein-Schatzman,2001; Kuchment-Zeng, 2001; E.-Post, 2003], theDirichlet case needed here is open (and difficult)

Quantum tunneling is neglected: recall that a truequantum-wire boundary is a finite potential jump

Drawbacks of these models

Presence of ad hoc parameters in the b.c. describingbranchings. A natural remedy: use a zero-width limit ina more realistic description

r−→

However, the answer is known so far only forNeumann-type situations [Rubinstein-Schatzman,2001; Kuchment-Zeng, 2001; E.-Post, 2003], theDirichlet case needed here is open (and difficult)

Quantum tunneling is neglected: recall that a truequantum-wire boundary is a finite potential jump

Leaky quantum graphs

We consider instead “leaky” graphs with an attractiveinteraction supported by graph edges. Formally we have

Hα,Γ = −∆− αδ(x− Γ) , α > 0 ,

in L2(Rn), where Γ is a graph in question, or generalizedgraph, understood as a subset of R

In this talk we will mostly consider the simplest graphs, orbuilding blocks or more complicated graphs, where Γ is asmooth manifold in R

n. We have in mind three cases:

curves in R2

surfaces in R3

curves in R3

Leaky quantum graphs

We consider instead “leaky” graphs with an attractiveinteraction supported by graph edges. Formally we have

Hα,Γ = −∆− αδ(x− Γ) , α > 0 ,

in L2(Rn), where Γ is a graph in question, or generalizedgraph, understood as a subset of R

In this talk we will mostly consider the simplest graphs, orbuilding blocks or more complicated graphs, where Γ is asmooth manifold in R

n. We have in mind three cases:

curves in R2

surfaces in R3

curves in R3

Definition of the HamiltonianIn the first two cases we have codim Γ = 1 and the operatorcan be defined by means of quadratic form,

ψ 7→ ‖∇ψ‖2L2(R2) − α

Γ|ψ(x)|2dx ,

which is closed and below bounded in W 2,1(R2); the secondterm makes sense in view of Sobolev embedding

For smooth manifolds and more general Γ such as a graphwith a locally finite number of smooth edges and no cuspswe can use an alternative definition by boundary conditions:Hα,Γ acts as −∆ on functions from W 1,2

loc (R2 \ Γ), which arecontinuous and exhibit a normal-derivative jump,

∂n(x)

−∂ψ

∂n(x)

= −αψ(x)

Definition of the HamiltonianIn the first two cases we have codim Γ = 1 and the operatorcan be defined by means of quadratic form,

ψ 7→ ‖∇ψ‖2L2(R2) − α

Γ|ψ(x)|2dx ,

which is closed and below bounded in W 2,1(R2); the secondterm makes sense in view of Sobolev embeddingFor smooth manifolds and more general Γ such as a graphwith a locally finite number of smooth edges and no cuspswe can use an alternative definition by boundary conditions:Hα,Γ acts as −∆ on functions from W 1,2

loc (R2 \ Γ), which arecontinuous and exhibit a normal-derivative jump,

∂n(x)

−∂ψ

∂n(x)

= −αψ(x)

The case codim Γ = 2

Boundary conditions can be used but they are morecomplicated. Moreover, for an infinite Γ corresponding toγ : R→ R

3 we have to assume in addition that there is atubular neighbourhood of Γ which does not intersect itself

Employ Frenet’s frame (t(s), b(s), n(s)) for Γ. Given ξ, η ∈ R

we set r = (ξ2+η2)1/2 and define family of “shifted” curves

Γr ≡ Γξηr := γr(s) ≡ γξηr (s) := γ(s) + ξb(s) + ηn(s)

Boundary conditions can be used but they are morecomplicated. Moreover, for an infinite Γ corresponding toγ : R→ R

3 we have to assume in addition that there is atubular neighbourhood of Γ which does not intersect itselfEmploy Frenet’s frame (t(s), b(s), n(s)) for Γ. Given ξ, η ∈ R

we set r = (ξ2+η2)1/2 and define family of “shifted” curves

Γr ≡ Γξηr := γr(s) ≡ γξηr (s) := γ(s) + ξb(s) + ηn(s)

The restriction of f ∈ W 2,2loc (R3 \ Γ) to Γr is well defined for

small r; we say that f ∈ W 2,2loc (R3 \Γ)∩L2(R3) belongs to Υ if

Ξ(f)(s) := − limr→0

ln rf Γr

Ω(f)(s) := limr→0

f Γr(s) + Ξ(f)(s) ln r

exist a.e. in R, are independent of the direction 1r (ξ, η), and

define functions from L2(R)

Then the operator Hα,Γ has the domain

g ∈ Υ : 2παΞ(g)(s) = Ω(g)(s)

and acts as

−Hα,Γf = −∆f for x ∈ R3 \ Γ

The restriction of f ∈ W 2,2loc (R3 \ Γ) to Γr is well defined for

small r; we say that f ∈ W 2,2loc (R3 \Γ)∩L2(R3) belongs to Υ if

Ξ(f)(s) := − limr→0

ln rf Γr

Ω(f)(s) := limr→0

f Γr(s) + Ξ(f)(s) ln r

exist a.e. in R, are independent of the direction 1r (ξ, η), and

define functions from L2(R)

Then the operator Hα,Γ has the domain

g ∈ Υ : 2παΞ(g)(s) = Ω(g)(s)

and acts as

−Hα,Γf = −∆f for x ∈ R3 \ Γ

RemarksIf Γ has components of codimension one and two, onecombines the above boundary conditions

The b.c. are natural describing point interaction in thenormal plane to Γ. Thus there is no way (at least withinstandard QM) to define Hα,Γ in the case codim Γ ≥ 4

Strong coupling considered below is closely related tosemiclassical behaviour of the operator

Hα,Γ(h) = −h2∆− αδ(x− Γ) , α > 0 ,

which can be regarded as h2Hα(h),Γ, where the effectivecoupling constant is α(h) := αh−2 for codim Γ = 1, and

α(h) := α +1

2πlnh if codim Γ = 2

Hα,Γ(h) = −h2∆− αδ(x− Γ) , α > 0 ,

α(h) := α +1

Hα,Γ(h) = −h2∆− αδ(x− Γ) , α > 0 ,

α(h) := α +1

Geometrically induced spectrumBending means binding, i.e. it may create isolatedeigenvalues of Hα,Γ. Consider a piecewise C1-smoothΓ : R→ R

2 parameterized by its arc length, and assume:

|Γ(s)− Γ(s′)| ≥ c|s− s′| holds for some c ∈ (0, 1)

Γ is asymptotically straight: there are d > 0, µ > 12

and ω ∈ (0, 1) such that

1−|Γ(s)− Γ(s′)|

|s− s′|≤ d

1 + |s+ s′|2µ]−1/2

in the sector Sω :=

(s, s′) : ω < ss′ < ω−1

straight line is excluded, i.e. |Γ(s)− Γ(s′)| < |s− s′|holds for some s, s′ ∈ R

Geometrically induced spectrumBending means binding, i.e. it may create isolatedeigenvalues of Hα,Γ. Consider a piecewise C1-smoothΓ : R→ R

2 parameterized by its arc length, and assume:

|Γ(s)− Γ(s′)| ≥ c|s− s′| holds for some c ∈ (0, 1)

Γ is asymptotically straight: there are d > 0, µ > 12

and ω ∈ (0, 1) such that

1−|Γ(s)− Γ(s′)|

|s− s′|≤ d

1 + |s+ s′|2µ]−1/2

in the sector Sω :=

(s, s′) : ω < ss′ < ω−1

straight line is excluded, i.e. |Γ(s)− Γ(s′)| < |s− s′|holds for some s, s′ ∈ R

Bending means binding

Theorem [E.-Ichinose, 2001]: Under these assumptions,σess(Hα,Γ) = [−1

4α2,∞) and Hα,Γ has at least one eigenvalue

below the threshold −14α

The same for curves in R3, under stronger regularity,

with −14α

2 is replaced by the corresponding 2D p.i. ev

For curved surfaces Γ ⊂ R3 such a result is proved in

the strong coupling asymptotic regime only

Implications for graphs: let Γ ⊃ Γ in the set sense, thenHα,Γ ≤ Hα,Γ. If the essential spectrum threshold is thesame for both graphs and Γ fits the above assumptions,we have σdisc(Hα,Γ) 6= ∅ by minimax principle

Bending means binding

Theorem [E.-Ichinose, 2001]: Under these assumptions,σess(Hα,Γ) = [−1

4α2,∞) and Hα,Γ has at least one eigenvalue

below the threshold −14α

The same for curves in R3, under stronger regularity,

with −14α

2 is replaced by the corresponding 2D p.i. ev

For curved surfaces Γ ⊂ R3 such a result is proved in

the strong coupling asymptotic regime only

Implications for graphs: let Γ ⊃ Γ in the set sense, thenHα,Γ ≤ Hα,Γ. If the essential spectrum threshold is thesame for both graphs and Γ fits the above assumptions,we have σdisc(Hα,Γ) 6= ∅ by minimax principle

Proof: generalized BS principleClassical Birman-Schwinger principle based on the identity

(H0 − V − z)−1 = (H0 − z)

−1 + (H0 − z)−1V 1/2

I − |V |1/2(H0 − z)−1V 1/2

−1|V |1/2(H0 − z)

can be extended to generalized Schrödinger operators Hα,Γ

[BEKŠ94]: the multiplication by (H0 − z)−1V 1/2 etc. is

replaced by suitable trace maps. In this way we find that−κ2 is an eigenvalue of Hα,Γ iff the integral operator Rκα,Γon L2(R) with the kernel

(s, s′) 7→α

κ|Γ(s)−Γ(s′)|)

has an eigenvalue equal to one

Sketch of the proofWe treat Rκα,Γ as a perturbation of the operator Rκα,Γ0

referring to a straight line. The spectrum of the latter isfound easily: it is purely ac and equal to [0, α/2κ)

The curvature-induced perturbation is sign-definite: wehave

Rκα,Γ −Rκα,Γ0

(s, s′) ≥ 0 , and the inequality is sharpsomewhere unless Γ is a straight line. Using a variationalargument with a suitable trial function we can check theinequality supσ(Rκα,Γ) > α

Due to the assumed asymptotic straightness of Γ theperturbation Rκα,Γ −R

κα,Γ0

is Hilbert-Schmidt , hence thespectrum of Rκα,Γ in the interval (α/2κ,∞) is discrete

Due to the assumed asymptotic straightness of Γ theperturbation Rκα,Γ −R

κα,Γ0

is Hilbert-Schmidt , hence thespectrum of Rκα,Γ in the interval (α/2κ,∞) is discrete

To conclude we employ continuity and limκ→∞ ‖Rκα,Γ‖ = 0.

The argument can be pictorially expressed as follows:Okayama University, March 24, 2004 – p.13/50

Pictorial sketch of the proof

pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppprpppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppr

σ(Rκα,Γ)

κα/2

Punctured manifoldsA natural question is what happens with σdisc(Hα,Γ) if Γ hasa small “hole”. We will give the answer for a compact,(n−1)-dimensional, C1+[n/2]-smooth manifold in R

Consider a family Sε0≤ε<η of subsets of Γ such that

each Sε is Lebesgue measurable on Γ

they shrink to origin, supx∈Sε|x| = O(ε) as ε→ 0

σdisc(Hα,Γ) 6= ∅, nontrivial for n ≥ 3

Punctured manifoldsA natural question is what happens with σdisc(Hα,Γ) if Γ hasa small “hole”. We will give the answer for a compact,(n−1)-dimensional, C1+[n/2]-smooth manifold in R

Consider a family Sε0≤ε<η of subsets of Γ such that

each Sε is Lebesgue measurable on Γ

they shrink to origin, supx∈Sε|x| = O(ε) as ε→ 0

σdisc(Hα,Γ) 6= ∅, nontrivial for n ≥ 3

Punctured manifolds: ev asymptotics

Call Hε := Hα,Γ\Sε. For small enough ε these operators

have the same finite number of eigenvalues, naturallyordered, which satisfy λj(ε)→ λj(0) as ε→ 0

Let ϕj be the eigenfunctions of H0. By Sobolev trace thmϕj(0) makes sense. Put sj := |ϕj(0)|

2 if λj(0) is simple,

otherwise they are ev’s of C :=(

ϕi(0)ϕj(0))

correspondingto a degenerate eigenvalue

Let ϕj be the eigenfunctions of H0. By Sobolev trace thmϕj(0) makes sense. Put sj := |ϕj(0)|

2 if λj(0) is simple,

otherwise they are ev’s of C :=(

ϕi(0)ϕj(0))

correspondingto a degenerate eigenvalue

Theorem [E.-Yoshitomi, 2003]: Under the assumptionsmade about the family Sε, we have

λj(ε) = λj(0) + αsjmΓ(Sε) + o(εn−1) as ε→ 0

Remarks

Formally a small-hole perturbation acts as a repulsive δinteraction with the coupling αmΓ(Sε)

No self-similarity of Sε required

If n = 2, i.e. Γ is a curve, mΓ(Sε) is the length of thehiatus. In this case the same asymptotic formula holdsfor bound states of an infinite curved Γ

Asymptotic perturbation theory for quadratic forms doesnot apply, because C∞

0 (Rn) 3 u 7→ |u(0)|2 ∈ R does notextend to a bounded form in W 1,2(Rn)

Remarks

Sketch of the proofTake an eigenvalue µ ≡ λj(0) of multiplicity m. It splits ingeneral, for small enough ε one has m eigenvalues insideC := z : |z − µ| < 3

4κ, where κ := 12dist (µ, σ(H0) \ µ)

&%'$r r q q qλj−1(0) µ C

Set wk(ζ, ε) := (Hε − ζ)−1ϕk − (H0 − ζ)

−1ϕk for ζ ∈ C andk = j, j + 1, . . . , j +m− 1. Using the choice of C andSobolev imbedding thm, one proves

‖wk(ζ, ε)‖W 1,2(Rn) = O(ε(n−1)/2) as ε→ 0

&%'$r r q q qλj−1(0) µ C

Set wk(ζ, ε) := (Hε − ζ)−1ϕk − (H0 − ζ)

−1ϕk for ζ ∈ C andk = j, j + 1, . . . , j +m− 1. Using the choice of C andSobolev imbedding thm, one proves

‖wk(ζ, ε)‖W 1,2(Rn) = O(ε(n−1)/2) as ε→ 0

Next, W 1,2(Rn) 3 f 7→ f |Γ ∈ L2(Γ) is compact; it implies

supζ∈C‖wk(ζ, ε)‖W 1,2(Rn) = o(ε(n−1)/2) as ε→ 0

Sketch of the proofLet Pε be spectral projection to these eigenvalues,

Pεϕk − ϕk =1

Cwk(ζ, ε) dζ = o(ε(n−1)/2)

in W 1,2(Rn) as ε→ 0

Pεϕk − ϕk =1

Cwk(ζ, ε) dζ = o(ε(n−1)/2)

Take m×m matrices L(ε) := ((HεPεϕi, Pεϕk)) andM(ε) := ((Pεϕi, Pεϕk)). We find that

((HεPεϕi, Pεϕk))− µδik − αϕi(0)ϕk(0)mΓ(Sε)

is o(εn−1) and (Pεϕi, Pεϕk) = δik + o(εn−1)

Pεϕk − ϕk =1

Cwk(ζ, ε) dζ = o(ε(n−1)/2)

Take m×m matrices L(ε) := ((HεPεϕi, Pεϕk)) andM(ε) := ((Pεϕi, Pεϕk)). We find that

((HεPεϕi, Pεϕk))− µδik − αϕi(0)ϕk(0)mΓ(Sε)

is o(εn−1) and (Pεϕi, Pεϕk) = δik + o(εn−1). Then

L(ε)M(ε)−1 = µI + αCmΓ(Sε) + o(εn−1)

and the claim of the theorem follows

Illustration: a ring with π20 cut

−10−5

R=6 α=1 θ=π/20 E0=−0.2535

Strong coupling for a compact Γ

Let Γ have a single component, smooth and compactTheorem [EY01, 02; EK03, Ex04]: (i) Let Γ be a C4 smoothmanifold. In the limit (−1)codimΓ−1α→∞ we have

#σdisc(Hα,Γ) =|Γ|α

2π+O(lnα)

for dim Γ = 1, codim Γ = 1,

2π+O(lnα)

#σdisc(Hα,Γ(h)) =|Γ|α2

16π2+O(lnα)

for dim Γ = 2, codim Γ = 1, and

2π+O(lnα)

#σdisc(Hα,Γ(h)) =|Γ|α2

16π2+O(lnα)

for dim Γ = 2, codim Γ = 1, and

#σdisc(Hα,Γ) =|Γ|(−εα)1/2

π+O(e−πα)

for dim Γ = 1, codim Γ = 2. Here |Γ| is the curve length orsurface area, respectively, and εα = −4 e2(−2πα+ψ(1))

Theorem, continued: (ii) In addition, suppose that Γ has noboundary . Then the j-th eigenvalue of Hα,Γ behaves as

λj(α) = −α2

4+ µj +O(α−1 lnα)

for codim Γ = 1 andλj(α) = εα + µj +O(eπα)

for codim Γ = 2,

where µj is the j-th eigenvalue of

SΓ = −d

ds2−

4k(s)2

on L2((0, |Γ|)) for dim Γ = 1, where k is curvature of Γ, andSΓ = −∆Γ +K −M2

on L2(Γ, dΓ) for dim Γ = 2, where −∆Γ is Laplace-Beltramioperator on Γ and K,M , respectively, are the correspondingGauss and mean curvatures

Theorem, continued: (ii) In addition, suppose that Γ has noboundary . Then the j-th eigenvalue of Hα,Γ behaves as

λj(α) = −α2

4+ µj +O(α−1 lnα)

for codim Γ = 1 andλj(α) = εα + µj +O(eπα)

for codim Γ = 2, where µj is the j-th eigenvalue of

SΓ = −d

ds2−

4k(s)2

on L2((0, |Γ|)) for dim Γ = 1, where k is curvature of Γ, andSΓ = −∆Γ +K −M2

on L2(Γ, dΓ) for dim Γ = 2, where −∆Γ is Laplace-Beltramioperator on Γ and K,M , respectively, are the correspondingGauss and mean curvatures

Proof techniqueConsider first the 1 + 1 case. Take a closed curve Γ and callL = |Γ|. We start from a tubular neighborhood of Γ

Lemma: Φa : [0, L)× (−a, a)→ R2 defined by

(s, u) 7→ (γ1(s)− uγ′2(s), γ2(s) + uγ′1(s)).

is a diffeomorphism for all a > 0 small enough

↑→

←→

Λouta

constant-width strip,do not take the LaTeXdrawing too literary!

Proof techniqueConsider first the 1 + 1 case. Take a closed curve Γ and callL = |Γ|. We start from a tubular neighborhood of Γ

Lemma: Φa : [0, L)× (−a, a)→ R2 defined by

(s, u) 7→ (γ1(s)− uγ′2(s), γ2(s) + uγ′1(s)).

is a diffeomorphism for all a > 0 small enough

↑→

←→

Λouta

constant-width strip,do not take the LaTeXdrawing too literary!

DN bracketingThe idea is to apply to the operator Hα,Γ in questionDirichlet-Neumann bracketing at the boundary ofΣa := Φ([0, L)× (−a, a)). This yields

(−∆NΛa

)⊕ L−a,α ≤ Hα,Γ ≤ (−∆D

Λa)⊕ L+

where Λa = Λina ∪ Λout

a is the exterior domain, and L±a,α are

self-adjoint operators associated with the forms

q±a,α[f ] = ‖∇f‖2L2(Σa) − α

Γ|f(x)|2 dS

where f ∈ W 1,20 (Σa) and W 1,2(Σa) for ±, respectively

Important : The exterior part does not contribute to thenegative spectrum, so we may consider L±

a,α only

DN bracketingThe idea is to apply to the operator Hα,Γ in questionDirichlet-Neumann bracketing at the boundary ofΣa := Φ([0, L)× (−a, a)). This yields

(−∆NΛa

)⊕ L−a,α ≤ Hα,Γ ≤ (−∆D

Λa)⊕ L+

where Λa = Λina ∪ Λout

a is the exterior domain, and L±a,α are

self-adjoint operators associated with the forms

q±a,α[f ] = ‖∇f‖2L2(Σa) − α

Γ|f(x)|2 dS

where f ∈ W 1,20 (Σa) and W 1,2(Σa) for ±, respectively

Important : The exterior part does not contribute to thenegative spectrum, so we may consider L±

a,α onlyOkayama University, March 24, 2004 – p.24/50

Transformed interior operatorWe use the curvilinear coordinates passing from L±

a,α tounitarily equivalent operators given by quadratic forms

b+a,α[f ] =

(1 + uk(s))−2

du ds +

−aV (s, u)|f |2 ds du− α

0|f(s, 0)|2 ds

with f ∈ W 1,2((0, L)× (−a, a)) satisfying periodic b.c. in thevariable s and Dirichlet b.c. at u = ±a, and

b−a,α[f ] = b+a,α[f ]−1∑

2(−1)j

1 + (−1)jak(s)|f(s, (−1)ja)|2 ds

where V is the curvature induced potential,

V (s, u) = −k(s)2

4(1+uk(s))2+

uk′′(s)

2(1+uk(s))3−

5u2k′(s)2

4(1+uk(s))4

Estimates with separated variablesWe pass to rougher bounds squeezing Hα,Γ between

H±a,α = U±

a ⊗ 1 + 1⊗ T±a,α

Here U±a are s-a operators on L2(0, L)

U±a = −(1∓ a‖k‖∞)−2 d2

ds2+ V±(s)

with PBC, where V−(s) ≤ V (s, u) ≤ V+(s) with an O(a) error,and the transverse operators are associated with the forms

t+a,α[f ] =

−a|f ′(u)|2 du− α|f(0)|2

t−a,α[f ] = t−a,α[f ]− ‖k‖∞(|f(a)|2 + |f(−a)|2)

with f ∈ W 1,20 (−a, a) and W 1,2(−a, a), respectively

Estimates with separated variablesWe pass to rougher bounds squeezing Hα,Γ between

H±a,α = U±

a ⊗ 1 + 1⊗ T±a,α

Here U±a are s-a operators on L2(0, L)

U±a = −(1∓ a‖k‖∞)−2 d2

ds2+ V±(s)

with PBC, where V−(s) ≤ V (s, u) ≤ V+(s) with an O(a) error,and the transverse operators are associated with the forms

t+a,α[f ] =

−a|f ′(u)|2 du− α|f(0)|2

t−a,α[f ] = t−a,α[f ]− ‖k‖∞(|f(a)|2 + |f(−a)|2)

with f ∈ W 1,20 (−a, a) and W 1,2(−a, a), respectively

Concluding the planar curve caseLemma: There are positive c, cN such that T±

α,a has for αlarge enough a single negative eigenvalue κ±α,a satisfying

−α2

1 + cNe−αa/2)

< κ−α,a < −α2

4< κ+

α,a < −α2

1− 8e−αa/2)

Finishing the proof:the eigenvalues of U±

a differ by O(a) from those of thecomparison operatorwe choose a = 6α−1 lnα as the neighbourhood widthputting the estimates together we get the eigenvalueasymptotics for a planar loop, i.e. the claim (ii)if Γ is not closed, the same can be done with thecomparison operators SD,N

Γ having appropriate b.c. atthe endpoints of Γ. This yields the claim (i)

Concluding the planar curve caseLemma: There are positive c, cN such that T±

α,a has for αlarge enough a single negative eigenvalue κ±α,a satisfying

−α2

1 + cNe−αa/2)

< κ−α,a < −α2

4< κ+

α,a < −α2

1− 8e−αa/2)

Finishing the proof:the eigenvalues of U±

a differ by O(a) from those of thecomparison operatorwe choose a = 6α−1 lnα as the neighbourhood widthputting the estimates together we get the eigenvalueasymptotics for a planar loop, i.e. the claim (ii)if Γ is not closed, the same can be done with thecomparison operators SD,N

Γ having appropriate b.c. atthe endpoints of Γ. This yields the claim (i)

A curve in R3

The argument is similar:

The “straightening” transformation Φa is defined by

Φa(s, r, θ) := γ(s)− r[n(s) cos(θ − β(s)) + b(s) sin(θ − β(s))]

To separate variables, we choose β so that β(s) equals thetorsion τ(s) of Γ. The effective potential is then

V = −k2

4h2+hss2h3−

where h := 1 + rk cos(θ − β). It is important that the leadingterm is −1

4k2 again, the torsion part being O(a)

A curve in R3

The argument is similar:

The “straightening” transformation Φa is defined by

Φa(s, r, θ) := γ(s)− r[n(s) cos(θ − β(s)) + b(s) sin(θ − β(s))]

To separate variables, we choose β so that β(s) equals thetorsion τ(s) of Γ. The effective potential is then

V = −k2

4h2+hss2h3−

where h := 1 + rk cos(θ − β). It is important that the leadingterm is −1

4k2 again, the torsion part being O(a)

A curve in R3

The transverse estimate is replaced byLemma: There are c1, c2 > 0 such that T±

α has for largeenough negative α a single negative ev κ±α,a which satisfies

εα − S(α) < κ−α,a < ξα < κ+α,a < ξα + S(α)

as α→ −∞, where S(α) = c1e−2πα exp(−c2e

−πα)

The rest of the argument is the same as above

Remark: Notice that the result extends easily to Γ’sconsisting of a finite number of connected components(curves) which are C4 and do not intersect. The same willbe true for surfaces considered below

A curve in R3

The transverse estimate is replaced byLemma: There are c1, c2 > 0 such that T±

α has for largeenough negative α a single negative ev κ±α,a which satisfies

εα − S(α) < κ−α,a < ξα < κ+α,a < ξα + S(α)

as α→ −∞, where S(α) = c1e−2πα exp(−c2e

−πα)

The rest of the argument is the same as above

Remark: Notice that the result extends easily to Γ’sconsisting of a finite number of connected components(curves) which are C4 and do not intersect. The same willbe true for surfaces considered below

A surface in R3

The argument modifies easily; Σa is now a layerneighborhood . However, the intrinsic geometry of Γcan no longer be neglected

Let Γ ⊂ R3 be a C4 smooth compact Riemann surface of a

finite genus g. The metric tensor given in the localcoordinates by gµν = p,µ · p,ν defines the invariant surfacearea element dΓ := g1/2d2s, where g := det(gµν).The Weingarten tensor is then obtained by raising the indexin the second fundamental form, hµ ν := −n,µ · p,σg

σν ; theeigenvalues k± of (hµ

ν) are the principal curvatures. Theydetermine Gauss curvature K and mean curvature M by

K = det(hµν) = k+k− , M =

2Tr (hµ

ν) =1

2(k++ k−)

A surface in R3

The argument modifies easily; Σa is now a layerneighborhood . However, the intrinsic geometry of Γcan no longer be neglectedLet Γ ⊂ R

3 be a C4 smooth compact Riemann surface of afinite genus g. The metric tensor given in the localcoordinates by gµν = p,µ · p,ν defines the invariant surfacearea element dΓ := g1/2d2s, where g := det(gµν).The Weingarten tensor is then obtained by raising the indexin the second fundamental form, hµ ν := −n,µ · p,σg

σν ; theeigenvalues k± of (hµ

ν) are the principal curvatures. Theydetermine Gauss curvature K and mean curvature M by

K = det(hµν) = k+k− , M =

2Tr (hµ

ν) =1

2(k++ k−)

Proof sketch in the surface case

The bracketing argument proceeds as before,

−∆NΛa⊕H−

α,Γ ≤ Hα,Γ ≤ −∆DΛa⊕H+

α,Γ , Λa := R3 \ Σa,

the interior only contributing to the negative spectrum

Using the curvilinear coordinates: For small enough a wehave the “straightening” diffeomorphism

La(x, u) = x+ un(x) , (x, u) ∈ Na := Γ× (−a, a)

Then we transform H±α,Γ by the unitary operator

Uψ = ψ La : L2(Ωa)→ L2(Na, dΩ)

and estimate the operators H±α,Γ := UH±

α,ΓU−1 in L2(Na, dΩ)

Proof sketch in the surface case

The bracketing argument proceeds as before,

−∆NΛa⊕H−

α,Γ ≤ Hα,Γ ≤ −∆DΛa⊕H+

α,Γ , Λa := R3 \ Σa,

the interior only contributing to the negative spectrumUsing the curvilinear coordinates: For small enough a wehave the “straightening” diffeomorphism

La(x, u) = x+ un(x) , (x, u) ∈ Na := Γ× (−a, a)

Then we transform H±α,Γ by the unitary operator

Uψ = ψ La : L2(Ωa)→ L2(Na, dΩ)

and estimate the operators H±α,Γ := UH±

α,ΓU−1 in L2(Na, dΩ)

Straightening transformationDenote the pull-back metric tensor by Gij,

(Gµν) 0

, Gµν = (δσµ − uhµσ)(δρσ − uhσ

ρ)gρν ,

so dΣ := G1/2d2s du with G := det(Gij) given by

G = g [(1− uk+)(1− uk−)]2 = g(1− 2Mu+Ku2)2

Let (·, ·)G denote the inner product in L2(Na, dΩ). Then H±α,Γ

are associated with the forms

η±α,Γ[U−1ψ] := (∂iψ,Gij∂jψ)G − α

Γ|ψ(s, 0)|2 dΓ ,

with the domains W 1,20 (Na, dΩ) and W 1,2(Na, dΩ) for the ±

sign, respectively

Straightening transformationDenote the pull-back metric tensor by Gij,

(Gµν) 0

, Gµν = (δσµ − uhµσ)(δρσ − uhσ

ρ)gρν ,

so dΣ := G1/2d2s du with G := det(Gij) given by

G = g [(1− uk+)(1− uk−)]2 = g(1− 2Mu+Ku2)2

Let (·, ·)G denote the inner product in L2(Na, dΩ). Then H±α,Γ

are associated with the forms

η±α,Γ[U−1ψ] := (∂iψ,Gij∂jψ)G − α

Γ|ψ(s, 0)|2 dΓ ,

with the domains W 1,20 (Na, dΩ) and W 1,2(Na, dΩ) for the ±

sign, respectivelyOkayama University, March 24, 2004 – p.32/50

Straightening continuedNext we remove 1− 2Mu+Ku2 from the weight G1/2 in theinner product of L2(Na, dΩ) by the unitary transformationU : L2(Na, dΩ)→ L2(Na, dΓdu),

Uψ := (1− 2Mu+Ku2)1/2ψ

Denote the inner product in L2(Na, dΓdu) by (·, ·)g. Theoperators B±

α,Γ := UH±α,ΓU

−1 are associated with the forms

b+α,Γ[ψ] = (∂µψ,Gµν∂νψ)g + (ψ, (V1 + V2)ψ)g

+‖∂uψ‖2g − α

Γ|ψ(s, 0)|2dΓ ,

b−α,Γ[ψ] = b+α,Γ[ψ] +1∑

(−1)j∫

ΓM(−1)ja(s)|ψ(s, (−1)ja)|2dΓ

for ψ from W 2,10 (Ωa, dΓdu) and W 2,1(Ωa, dΓdu), respectively

Straightening continuedNext we remove 1− 2Mu+Ku2 from the weight G1/2 in theinner product of L2(Na, dΩ) by the unitary transformationU : L2(Na, dΩ)→ L2(Na, dΓdu),

Uψ := (1− 2Mu+Ku2)1/2ψ

Denote the inner product in L2(Na, dΓdu) by (·, ·)g. Theoperators B±

α,Γ := UH±α,ΓU

−1 are associated with the forms

b+α,Γ[ψ] = (∂µψ,Gµν∂νψ)g + (ψ, (V1 + V2)ψ)g

+‖∂uψ‖2g − α

Γ|ψ(s, 0)|2dΓ ,

b−α,Γ[ψ] = b+α,Γ[ψ] +1∑

(−1)j∫

ΓM(−1)ja(s)|ψ(s, (−1)ja)|2dΓ

for ψ from W 2,10 (Ωa, dΓdu) and W 2,1(Ωa, dΓdu), respectively

Effective potential

Here Mu := (M −Ku)(1− 2Mu+Ku2)−1 is the meancurvature of the parallel surface to Γ and

V1 = g−1/2(g1/2GµνJ,ν),µ+J,µGµνJ,ν , V2 =

K −M2

(1− 2Mu+Ku2)2

with J := 12 ln(1− 2Mu+Ku2)

Effective potential

K −M2

(1− 2Mu+Ku2)2

A rougher estimate with separated variables: squeeze1− 2Mu+Ku2 between C±(a) := (1± a%−1)2, where% := max(‖k+‖∞ , ‖k−‖∞)

−1. Consequently, the matrixinequality C−(a)gµν ≤ Gµν ≤ C+(a)gµν is valid

Effective potential

K −M2

(1− 2Mu+Ku2)2

A rougher estimate with separated variables: squeeze1− 2Mu+Ku2 between C±(a) := (1± a%−1)2, where% := max(‖k+‖∞ , ‖k−‖∞)

−1. Consequently, the matrixinequality C−(a)gµν ≤ Gµν ≤ C+(a)gµν is valid

V1 behaves as O(a) for a→ 0, while V2 can be squeezedbetween the functions C−2

± (a)(K −M2), both uniformly inthe surface variables

Concluding the estimateHence we estimate B±

α,Γ by

B±α,a := S±

a ⊗ I + I ⊗ T±α,a

with S±a := −C±(a)∆Γ + C−2

± (a)(K −M2)± va in the spaceL2(Γ, dΓ)⊗L2(−a, a) for a v > 0, where T±

α,a are the same asin the 1 + 1 case (the same lemma applies)

α,Γ by

B±α,a := S±

a ⊗ I + I ⊗ T±α,a

with S±a := −C±(a)∆Γ + C−2

α,a are the same asin the 1 + 1 case (the same lemma applies)As above the eigenvalues of the operators S±

a coincide upto an O(a) error with those of SΓ, and therefore choosinga := 6α−1 lnα, we find

λj(α) = −1

4α2 + µj +O(α−1 lnα)

as a→ 0 which is equivalent to the claim (i)

α,Γ by

B±α,a := S±

a ⊗ I + I ⊗ T±α,a

with S±a := −C±(a)∆Γ + C−2

α,a are the same asin the 1 + 1 case (the same lemma applies)As above the eigenvalues of the operators S±

a coincide upto an O(a) error with those of SΓ, and therefore choosinga := 6α−1 lnα, we find

λj(α) = −1

4α2 + µj +O(α−1 lnα)

as a→ 0 which is equivalent to the claim (i)To get (ii) we employ Weyl asymptotics for SΓ. Extension toΓ’s having a finite # of connected components is easy

Infinite manifoldsBound states may exist also if Γ is noncompact . Thecomparison operator SΓ has an attractive potential, soσdisc(Hα,Γ) 6= ∅ can be expected in the strong couplingregime, even if a direct proof is missing as for surfaces

It is needed that σess does not feel curvature, not only forHα,Γ but for the estimating operators as well. Sufficientconditions:

k(s), k′(s) and k′′(s)1/2 are O(|s|−1−ε) as |s| → ∞for a planar curve

in addition, the torsion bounded for a curve in R3

a surface Γ admits a global normal parametrization witha uniformly elliptic metric, K,M → 0 as the geodesicradius r →∞

Infinite manifoldsBound states may exist also if Γ is noncompact . Thecomparison operator SΓ has an attractive potential, soσdisc(Hα,Γ) 6= ∅ can be expected in the strong couplingregime, even if a direct proof is missing as for surfaces

It is needed that σess does not feel curvature, not only forHα,Γ but for the estimating operators as well. Sufficientconditions:

k(s), k′(s) and k′′(s)1/2 are O(|s|−1−ε) as |s| → ∞for a planar curve

in addition, the torsion bounded for a curve in R3

a surface Γ admits a global normal parametrization witha uniformly elliptic metric, K,M → 0 as the geodesicradius r →∞

Infinite manifolds

We must also assume that there is a tubular neighborhoodΣa without self-intersections for small a, i.e. to avoid

Theorem [EY02; EK03, Ex04]: With the above listedassumptions, the asymptotic expansions (ii) for theeigenvalues derived in the compact case hold again

Infinite manifolds

We must also assume that there is a tubular neighborhoodΣa without self-intersections for small a, i.e. to avoid

Theorem [EY02; EK03, Ex04]: With the above listedassumptions, the asymptotic expansions (ii) for theeigenvalues derived in the compact case hold again

Periodic manifoldsOne uses Floquet expansion. It is important to choose theperiodic cells C of the space and ΓC of the manifoldconsistently, ΓC = Γ ∩ C; we assume that ΓC is connected

Lemma: ∃ unitary U : L2(R3)→∫ ⊕[0,2π)r L2(C) dθ s.t.

UHα,ΓU−1 =

∫ ⊕

[0,2π)r

Hα,θ dθ and σ(Hα,Γ)=⋃

[0,2π)r

σ(Hα,θ)

Periodic manifoldsOne uses Floquet expansion. It is important to choose theperiodic cells C of the space and ΓC of the manifoldconsistently, ΓC = Γ ∩ C; we assume that ΓC is connected

Lemma: ∃ unitary U : L2(R3)→∫ ⊕[0,2π)r L2(C) dθ s.t.

UHα,ΓU−1 =

∫ ⊕

[0,2π)r

Hα,θ dθ and σ(Hα,Γ)=⋃

[0,2π)r

σ(Hα,θ)

Comparison operators

The fibre comparison operators are

Sθ = −d

ds2−

4k(s)2

on L2(ΓC) parameterized by arc length for dim Γ = 1, withFloquet b.c., and

Sθ = g−1/2(−i∂µ + θµ)g1/2gµν(−i∂ν + θν) +K −M2

with periodic b.c. for dim Γ = 2, where θµ, µ = 1, . . . , r,are quasimomentum components; recall that r = 1, 2, 3depending on the manifold type

Periodic manifold asymptotics

Theorem [EY01; EK03, Ex04]: Let Γ be a C4-smoothr-periodic manifold without boundary. The strong couplingasymptotic behavior of the j-th Floquet eigenvalue is

λj(α, θ) = −1

4α2 + µj(θ) +O(α−1 lnα) as α→∞

for codim Γ = 1 and

λj(α, θ) = εα + µj(θ) +O(eπα) as α→ −∞

for codim Γ = 2. The error terms are uniform w.r.t. θ

Corollary: If dim Γ = 1 and coupling is strong enough,Hα,Γ has open spectral gaps

Periodic manifold asymptotics

Theorem [EY01; EK03, Ex04]: Let Γ be a C4-smoothr-periodic manifold without boundary. The strong couplingasymptotic behavior of the j-th Floquet eigenvalue is

λj(α, θ) = −1

4α2 + µj(θ) +O(α−1 lnα) as α→∞

for codim Γ = 1 and

λj(α, θ) = εα + µj(θ) +O(eπα) as α→ −∞

for codim Γ = 2. The error terms are uniform w.r.t. θ

Corollary: If dim Γ = 1 and coupling is strong enough,Hα,Γ has open spectral gaps

Large gaps in the disconnected case

If Γ is not connected and each connected component iscontained in a translate of ΓC, the comparison operator isindependent of θ and asymptotic formula reads

λj(α, θ) = −1

4α2 + µj +O(α−1 lnα) as α→∞

for codim Γ = 1 and similarly for for codim Γ = 2

Moreover, the assumptions can be weakened

Large gaps in the disconnected case

If Γ is not connected and each connected component iscontained in a translate of ΓC, the comparison operator isindependent of θ and asymptotic formula reads

λj(α, θ) = −1

4α2 + µj +O(α−1 lnα) as α→∞

for codim Γ = 1 and similarly for for codim Γ = 2

Moreover, the assumptions can be weakened

Soft graphs with magnetic fieldAdd a homogeneous magnetic field with the vector potentialA = 1

2B(−x2, x1). We ask about existence of persistentcurrents, i.e. nonzero probability flux along a closed loop

∂λn(φ)

∂φ= −

where λn(φ) is the n-th eigenvalue of the Hamiltonian

Hα,Γ(B) := (−i∇− A)2 − αδ(x− Γ)

and φ is the magnetic flux through the loop (in standardunits its quantum equals 2π~c|e|−1)

Soft graphs with magnetic fieldAdd a homogeneous magnetic field with the vector potentialA = 1

2B(−x2, x1). We ask about existence of persistentcurrents, i.e. nonzero probability flux along a closed loop

∂λn(φ)

∂φ= −

where λn(φ) is the n-th eigenvalue of the Hamiltonian

Hα,Γ(B) := (−i∇− A)2 − αδ(x− Γ)

and φ is the magnetic flux through the loop (in standardunits its quantum equals 2π~c|e|−1)

Persistent currentsThe same technique, different comparison operator, namely

SΓ(B) = −d

ds2−

4k(s)2

on L2(0, L) with ψ(L−) = eiB|Ω|ψ(0+), ψ′(L−) = eiB|Ω|ψ′(0+),where Ω is the area encircled by Γ

SΓ(B) = −d

ds2−

4k(s)2

Theorem [E.-Yoshitomi, 2003]: Let Γ be a C4-smooth. Thefor large α the operator Hα,Γ(B) has a non-empty discretespectrum and the j-th eigenvalue behaves as

λj(α,B) = −1

4α2 + µj(B) +O(α−1 lnα) ,

where µj(B) is the j-th eigenvalue of SΓ(B) and the errorterm is uniform in B. In particular, for a fixed j and α largeenough the function λj(α, ·) cannot be constant

SΓ(B) = −d

ds2−

4k(s)2

Theorem [E.-Yoshitomi, 2003]: Let Γ be a C4-smooth. Thefor large α the operator Hα,Γ(B) has a non-empty discretespectrum and the j-th eigenvalue behaves as

λj(α,B) = −1

4α2 + µj(B) +O(α−1 lnα) ,

where µj(B) is the j-th eigenvalue of SΓ(B) and the errorterm is uniform in B. In particular, for a fixed j and α largeenough the function λj(α, ·) cannot be constantRemark: [Honnouvo-Hounkonnou, 2004] proved the same for AB flux

Absolute continuity

One is also interested in the nature of the spectrum of Hα,Γ

with a periodic Γ. By [Birman-Suslina-Shterenberg 00, 01]the spectrum is absolutely continuous if codim Γ = 1 and theperiod cell is compact. This tells us nothing, e.g., about asingle periodic curve in R

d, d = 2, 3.

The assumption about connectedness of ΓC can be alwayssatisfied if d = 2 but not for d = 3; recall the crotchet curve

Absolute continuity

One is also interested in the nature of the spectrum of Hα,Γ

with a periodic Γ. By [Birman-Suslina-Shterenberg 00, 01]the spectrum is absolutely continuous if codim Γ = 1 and theperiod cell is compact. This tells us nothing, e.g., about asingle periodic curve in R

d, d = 2, 3.

The assumption about connectedness of ΓC can be alwayssatisfied if d = 2 but not for d = 3; recall the crotchet curve

Absolute continuity

Theorem [Bentosela-Duclos-E., 2003]: To any E > 0 thereis an αE > 0 such that the spectrum of Hα,Γ is absolutelycontinuous in (−∞, ξ(α) + E) as long as (−1)dα > αE,where ξ(α) = −1

4α2 and εα for d = 2, 3, respectively

Proof: The fiber operators Hα,Γ(θ) form a type A analyticfamily. In a finite interval each of them has a finite numberof ev’s , so one has just to check non-constancy of thefunctions λj(α, ·) as in the case of persistent currents

Absolute continuity

Theorem [Bentosela-Duclos-E., 2003]: To any E > 0 thereis an αE > 0 such that the spectrum of Hα,Γ is absolutelycontinuous in (−∞, ξ(α) + E) as long as (−1)dα > αE,where ξ(α) = −1

4α2 and εα for d = 2, 3, respectively

Proof: The fiber operators Hα,Γ(θ) form a type A analyticfamily. In a finite interval each of them has a finite numberof ev’s , so one has just to check non-constancy of thefunctions λj(α, ·) as in the case of persistent currents

Open questionsStrong coupling, manifolds with boundary: If Γ has aboundary, we have a strong-coupling asymptotics forthe bound state number but not for ev’s themselves.We conjecture that the latter is given again by

λj(α) = −1

4α2 + µj +O(α−1 lnα) ,

etc., where µj refers to SΓ with Dirichlet b.c.

Strong coupling, less regularity: Examples show thatthe above relation is not valid for a non-smooth Γ, ratherµj is replaced by a term proportional to α2. How doesthe asymptotic expansion look in this case and how itdepends on dimension and codimension of Γ? Theanalogous question can be asked more generally forgraphs with branching points and generalized graphs

Open questionsStrong coupling, manifolds with boundary: If Γ has aboundary, we have a strong-coupling asymptotics forthe bound state number but not for ev’s themselves.We conjecture that the latter is given again by

λj(α) = −1

4α2 + µj +O(α−1 lnα) ,

etc., where µj refers to SΓ with Dirichlet b.c.

Strong coupling, less regularity: Examples show thatthe above relation is not valid for a non-smooth Γ, ratherµj is replaced by a term proportional to α2. How doesthe asymptotic expansion look in this case and how itdepends on dimension and codimension of Γ? Theanalogous question can be asked more generally forgraphs with branching points and generalized graphs

Open questions

Scattering theory on non-compact “leaky” curves,manifolds, graphs, and generalized graphs:

existence and completeness, including spectral a.c.in (−1

4α2, 0) w.r.t. asymptotic geometry of Γ

asymptotic behavior of S-matrix in strong-couplingregime, including relations between S-matrices ofleaky and “ideal” graphsprove existence of resonances, at least withinparticular models

Periodic Γ: one expects that the whole spectrum isabsolutely continuous, independently of α, but itremains to be proved. Also strong-coupling asymptoticproperties of spectral gaps are not known

Open questions

asymptotic behavior of S-matrix in strong-couplingregime, including relations between S-matrices ofleaky and “ideal” graphs

prove existence of resonances, at least withinparticular models

Open questions

Random graphs, either by their shape or by a randomcoupling α : Γ→ R+. Is it true that the whole negativepart of σess(Hα,Γ) is always p.p. once a disorder ispresent?

Adding magnetic field: Will the curvature-induceddiscrete spectrum survive under any magnetic field?On the other hand, will (at least a part of) the a.c.spectrum of (−i∇− A)2 − αδ(x− Γ) survive arandomization of a straight Γ?

etc, etc

Open questions

etc, etc

Open questions

etc, etc

The talk was based on[BDE03] F. Bentosela, P. Duclos, P.E.: Absolute continuity in periodic thin tubes and

strongly coupled leaky wires, Lett. Math. Phys. 65 (2003), 75-82.[BEKŠ94] J.F. Brasche, P.E., Yu.A. Kuperin, P. Šeba: Schrödinger operators with singular

interactions, J. Math. Anal. Appl. 184 (1994), 112–139.[Ex01] P.E.: Bound states of infinite curved polymer chains, Lett. Math. Phys. 57 (2001),

87-96.[Ex04] P.E.: Spectral properties of Schrödinger operators with a strongly attractive δ

interaction supported by a surface, to appear in Proceedings of the NSF SummerResearch Conference (Mt. Holyoke 2002); AMS “Contemporary Mathematics" Series,Providence, R.I., 2004

[EI01] P.E., T. Ichinose: Geometrically induced spectrum in curved leaky wires, J. Phys.A34 ( 2001), 1439-1450.

[EK02] P.E., S. Kondej: Curvature-induced bound states for a δ interaction supported by acurve in R

3, Ann. H. Poincaré 3 (2002), 967-981.[EK03a] P.E., S. Kondej: Bound states due to a strong δ interaction supported by a curved

surface, J. Phys. A36 (2003), 443-457.[EK03b] P.E., S. Kondej: Strong-coupling asymptotic expansion for Schrödinger operators

with a singular interaction supported by a curve in R3, Rev. math. Phys., to appear;

math-ph/0303033

And it is not all, see also

[EK03c] P.E., S. Kondej: Schrödinger operators with singular interactions: a model oftunneling resonances, math-ph/0312055

[EN03] P.E., K. Nemcová: Leaky quantum graphs: approximations by point interactionHamiltonians, J. Phys. A36 (2003), 10173-10193.

[ET04] P.E., M. Tater: Spectra of soft ring graphs, Waves in Random Media 13 (2003),S47-S60.

[EY01] P.E., K. Yoshitomi: Band gap of the Schrödinger operator with a strong δ-interactionon a periodic curve, Ann. H. Poincaré 2 (2001), 1139-1158.

[EY02a] P.E., K. Yoshitomi: Asymptotics of eigenvalues of the Schrödinger operator with astrong δ-interaction on a loop, J. Geom. Phys. 41 (2002), 344-358.

[EY02b] P.E., K. Yoshitomi: Persistent currents for 2D Schrödinger operator with a strongδ-interaction on a loop, J. Phys. A35 (2002), 3479-3487.

[EY03] P.E., K. Yoshitomi: Eigenvalue asymptotics for the Schrödinger operator with aδ-interaction on a punctured surface, Lett. Math. Phys. 65 (2003), 19-26.

Fortunately, you need not copy all of this – to find links tothese papers see http://www.ujf.cas.cz/ exner